Nous construisons une action algébrique non linéarisable (i.e. pas conjuguée à une action linéaire) du groupe des permutations de 3 éléments sur l’espace affine complexe de dimension quatre. Plus généralement, cette action peut être utilisée pour construire des actions non linéarisables de sur pour tout entier .
The main purpose of this article is to give an explicit algebraic action of the group of permutations of 3 elements on affine four-dimensional complex space which is not conjugate to a linear action.
@article{AIF_2002__52_1_133_0, author = {Freudenburg, Gene and Moser-Jauslin, Lucy}, title = {A nonlinearizable action of $S\_3$ on ${\mathbb {C}}^4$}, journal = {Annales de l'Institut Fourier}, volume = {52}, year = {2002}, pages = {133-143}, doi = {10.5802/aif.1879}, mrnumber = {1881573}, zbl = {1028.14019}, language = {en}, url = {http://dml.mathdoc.fr/item/AIF_2002__52_1_133_0} }
Freudenburg, Gene; Moser-Jauslin, Lucy. A nonlinearizable action of $S_3$ on ${\mathbb {C}}^4$. Annales de l'Institut Fourier, Tome 52 (2002) pp. 133-143. doi : 10.5802/aif.1879. http://gdmltest.u-ga.fr/item/AIF_2002__52_1_133_0/
[A] Non-linearizable algebraic -actions on affine spaces, Invent. Math., Tome 138 (1999) no. 2, pp. 281-306 | Article | MR 1720185 | Zbl 0933.14027
[BH1] Linearizing certain reductive group actions, Trans. Amer. Math. Soc., Tome 292 (1985), pp. 463-482 | Article | MR 808732 | Zbl 0602.14047
[BH2] Some equivariant -theory of affine algebraic group actions, Comm. Algebra, Tome 15 (1987), pp. 181-217 | Article | MR 876977 | Zbl 0612.14047
[DK] Non-linearizable holomorphic group actions, Math. Annalen, Tome 311 (1998), pp. 41-53 | Article | MR 1624259 | Zbl 0911.32042
[HK] Le principe d'Oka équivariant, C.R. Acad. Sci. Paris, Série I, Tome 315 (1992), pp. 217-220 | MR 1194532 | Zbl 0782.32021
[Kr1] Algebraic automorphisms of affine space, Topological Methods in Algebraic Transformation Groups, Proceedings of a Conference at Rutgers, Birkhäuser (Progress in Mathematics) Tome Vol. 80 (1989), pp. 81-106 | Zbl 0719.14030
[Kr2] G-vector bundles and the linearization problem, Group Actions and Invariant Theory, Proceeding of the 1988 Montreal Conference, CMS Conference Proceedings (Canadian Math. Soc.) Tome Vol. 10 (1988), pp. 111-124 | Zbl 0703.14009
[KS] Reductive group actions with one dimensional quotient, Publ. Math. IHES, Tome 76 (1992), pp. 1-97 | Numdam | MR 1215592 | Zbl 0783.14026
[Med] Moduli of -equivariant vector bundles (1995) (Thesis, Brandeis Univ.)
[MJ] Triviality of certain equivariant vector bundles for finite cyclic groups, C.R. Acad. Sci. Paris, Tome 317 (1993), pp. 139-144 | MR 1231410 | Zbl 0813.14033
[MMP1] Equivariant algebraic vector bundles over cones with smooth one dimensional quotient, J. Math. Soc. Japan, Tome 50 (1998), pp. 379-414 | Article | MR 1613156 | Zbl 0928.14013
[MMP2] The equivariant Serre problem for abelian groups, Topology, Tome 35 (1996), pp. 329-334 | Article | MR 1380501 | Zbl 0884.14007
[MP] Stably trivial equivariant algebraic vector bundles, J. Amer. Math. Soc., Tome 8 (1995), pp. 687-714 | Article | MR 1303027 | Zbl 0862.14009
[Q] Projective modules over polynomial rings, Invent. Math., Tome 36 (1976), pp. 167-171 | Article | MR 427303 | Zbl 0337.13011
[Sch] Exotic algebraic group actions, C.R. Acad. Sci. Paris, Tome 309 (1989), pp. 89-94 | MR 1004947 | Zbl 0688.14040
[Sus] Projective modules over a polynomial ring, Soviet Doklady, Tome 17 (1976), pp. 1160-1164 | Zbl 0354.13010
[Sus] Projective modules over a polynomial ring, Dokl. Acad. Nauk. SSSR, Tome 26 (1976)